Structure of a steady drain-hole vortex in a viscous fluid

We use direct numerical simulations to study a steady bathtub vortex in a cylindrical tank with a central drain-hole, a fiat stress-free surface and velocity prescribed at the inlet. We find that the qualitative structure of the meridional flow does not depend on the radial Reynolds number, whereas we observe a weak overall rotation at a low radial Reynolds number and a concentrated vortex above the drain-hole at a high radial Reynolds number. We introduce a simple analytically integrable model that shows the same qualitative dependence on the radial Reynolds number as the simulations and compares favourably with the results for the radial velocity and the azimuthal velocity at the surface. Finally, we describe the height dependence of the radius of the vortex core and the maximum of the azimuthal velocity at a high radial Reynolds number, and we show that the data on the radius of the vortex core and the maximum of the azimuthal velocity as functions of height collapse on single curves by appropriate scaling.